MAS295 Vector Spaces (NJTech)
|Semester 2, 2016/17||10 Credits|
|Lecturer:||Dr Fionntan Roukema||Reading List|
This course introduces abstract vector spaces and linear transformations, building on the concrete approach with matrices in earlier courses. Many results that were merely stated, or proved by matrix methods, will be given conceptual proofs in this course. The abstract approach will allow us to give efficient proofs that simultaneously tell us interesting things about vectors, matrices, polynomials, sequences, differential equations, and many other objects. A central aim of the course is to help students become comfortable with the required level of abstraction.
There are no prerequisites for this module.
No other modules have this module as a prerequisite.
- Vector spaces, linear maps, subspaces.
- Independence and spanning sets.
- Linear maps out of Rn; matrices for linear maps.
- Theorems about bases.
- Eigenvalues and eigenvectors.
- Introduce the abstract theory of vector spaces and linear maps between them.
- Introduce the abstract theory of inner product spaces.
- Familiarise students with abstract and axiomatic mathematics.
- Students should understand the main concepts of linear algebra: vector spaces; subspaces, (direct) sum and intersection; linear maps, kernels and images; independent sets, spanning sets and bases; relations with matrices.
- Students should be familiar with a range of examples of these concepts.
- Students should be able to prove the main theorems about these concepts.
16 lectures, 16 tutorials
One formal 2 hour written examination. All questions compulsory.
1. Vector spaces
Definitions and examples.
Definitions and examples. 3. Subspaces
Definitions and examples, (direct) sums and intersections. 4. Independence and spanning sets
Definitions and examples. Bases. 5. Linear maps out of Rn
A linear map Rn→ V is the same as a list of n elements of V. 6. Matrices for linear maps
Definitions, properties, and behaviour under change of basis. 7. Determinant and trace
The definition of determinant and trace for linear operators on a finite dimensional space. 8. Theorems about bases
Invariance of dimension, rank-nullity formula, dim(U+V)+dim(U∩V)=dim(U)+dim(V). 9. Eigenvalues and eigenvectors
A brief account for abstract vector spaces.
|C||Friedberg, Insel, Spence||Linear Algebra||Blackwells||Amazon|
|C||Halmos||Finite-dimensional vector spaces||Blackwells||Amazon|
(A = essential, B = recommended, C = background.)
Most books on reading lists should also be available from the Blackwells shop on Mappin Street.